An Efficient Genetic Algorithm for the Power-Balanced Anycast Routing Problem in Static Ad Hoc Networks

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An Efficient Genetic Algorithm for the Power-Balanced Anycast

Routing Problem in Static Ad Hoc Networks*

Pi-Rong Sheu and Yu-Ting Li

Department of Electrical Engineering

National Yunlin University of Science & Technology

Touliu, Yunlin 640, Taiwan, R.O.C.

Email: [email protected]

Abstract

In this paper, we will consider the problem of designing an anycast routing algorithm to efficiently arrange anycast transmission requirements such that the power consumption of each node is as even as possible. In this paper, we will prove the problem to be an NP-complete problem and use the principle of genetic algorithms to design a novel heuristic algorithm for the difficult problem. Computer simulations verify that the power consumptions of networks generated by our genetic power-balanced anycast routing algorithm are more balanced than those generated by shortest-path-based anycast routing algorithms and other traditional power-balanced anycast routing algorithms.

Keywords: Ad Hoc Network, Anycast Routing,

NP-Completeness, Genetic Algorithm, Power-Aware, Power-Balanced.

摘要

在本論文中我們將考慮如何設計一個 anycast 路徑繞送演算法來有效地安排 anycast 之傳送需求以使得每一個節點的電力消耗能盡可能地平均。在本論文中，我們將證明如此之電力平衡anycast 路徑繞送問題為一個 NP-complete 的問題並為此問題設計一個以基因演算法為主之啟發式演算法。電腦模擬結果顯示我們的電力平衡 anycast 路徑繞送之基因演算法所產生的網路生命週期將較長於以最短路徑為主之 anycast 路徑繞送演算法以及其它傳統之電力平衡 anycast 路徑繞送演算法。 關鍵字: 隨意網路，Anycast 路徑繞送，NP 完全性， 基因演算法，電力平衡，電力考量

1. Introduction

Battery power has always remained one of the central issues in ad hoc networks. This is because the operation of a host in an ad hoc network is totally subject to its power capacity and consumption rate [7]. When battery power is drained, the host will disappear from the ad hoc network, risking the overall operation of the network as well as the transmissions of data packets. In designing routing protocols for ad hoc networks, if the factor of node’s power consumption is neglected, two undesirable consequences may arise. First, every node *This work was supported by the National Science Council of the Republic of China under Grant # NSC 93-2213-E-224-023

may experience an unequal degree of power consumption. As a result, some nodes will consume power faster than other nodes. Evidently, the lifetime of network will be shortened [7, 9]. Second, the overall power in the network will be consumed on a large scale, endangering its lifetime (One of the common definitions of the network’s lifetime is the time period from the beginning of the network’s operation to the time when one of the nodes exhausts its battery power [5].) In view of these flaws, the inclusion of a power-aware mechanism into routing protocols (or algorithms) has recently become a focus of study in an ad hoc network. In general, there are two main strategies for designing power-aware routing protocols (or algorithms) in the related literature. The first strategy attempts to reduce each node’s power consumption equally such that the lifetime of network is prolonged. The other tries to decrease the network’s overall battery power consumption in quest of a longer network’s lifetime. In this paper, we will take the approach of leveling each node’s power consumptions as a starting point.

In ad hoc networks , common communication models among hosts include unicast (one-to-one) transmission, multicast (one-to-many) transmission, and broadcast (one-to-all) transmission. Besides, anycast (one-to-any) transmission has been comprehensively investigated and become an important communication model recently [8]. For instance, there has existed a special field in an IPv6 packet which is dedicated to the setting of anycast transmission. In this paper, we are mainly concerned with anycast transmission. In anycast transmission, a number of servers (or called source nodes) can be reached by an identical anycast address and provide the same type of services and resources. When a client (or called a destination node) wants to get services or resources, the anycast routing algorithm will arrange the most suitable server to it. For example, to provide the outcome checking service for the examinees, we can evenly install various database hosts (called source hosts) in an ad hoc network. Whenever a user host (called destination host) makes the request of checking service, the anycast routing algorithm will automatically order the source host which is closest to this destination host to provide its checking service.

When designing anycast routing algorithms for ad hoc networks, we must take the main characteristic of ad hoc networks into consideration: the battery power of

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each host is very limited. If the routing requirements are arranged by those anycast routing algorithms only armed with shortest-path routing paths, individual power consumption may vary although the total transmission power consumption is smaller. As a result, the network may suffer very short lifetime. Instead, if we attempt to evenly distribute packet-relaying loads among nodes to prevent the overuse or abuse of battery power, we believe that the network’s lifetime will be extended significantly. For example, in the above examination system, when the requirements of most clients are satisfied by the same server, the battery power of some nodes (including source nodes and intermediate nodes) may be consumed abnormally quickly, which implies the lifetime of network may be shortened significantly.

In this paper we will thus define our goal of study: designing an efficient anycast routing algorithm to arrange anycast transmission requirements such that the power consumption of each node is as balanced as possible. We will call it the power-balanced anycast

routing (PBAR) problem. To be more specific, given a set

of destination nodes each of which requires a different amount of data packets, find a set of routing paths between source nodes and destination nodes such that the power consumption of each node is as even as possible. Undoubted, the resulted lifetime of network is extended. While it is easy to find a set of routing paths with the minimal total transmission power consumption to satisfy the data packet requirement of each destination hosts, in this paper we will prove that the PBAR problem is an NP-hard problem.

In 1975, John Holland proposed that the evolutionary processes found in nature may be used to solve complex optimization problem [4]. Such a computing paradigm, commonly known as Genetic Algorithm (GA), has been successfully employed in solving a variety of problems. Especially, GAs have successfully been applied to a number of NP-complete problems, including many famous NP-complete problems in communication networks, such as the multicast routing problem. Recently, many researchers have attempted to adopt GAs to solve various problems existing in ad hoc networks [1]. As a result, it is worthy to develop efficient GA to yield the better solutions for the PBAR problem. In this paper, we will use the principle of GAs to design a novel heuristic algorithm for the PBAR problem. Computer simulations verify that the power consumptions of networks generated by our genetic power-balanced anycast routing algorithm are more balanced than those generated by shortest-path-based anycast routing algorithms and other traditional power-balanced anycast routing algorithms.

The rest of the paper is organized as follows. In Section 2, a formal definition of our PBAR problem is given. In Section 3, our PBAR problem is proved to be an NP-complete problem. In Section 4, an efficient GA for the PBAR problem is proposed. In Section 5, the performance of our GA is evaluated through simulations

and compared to other heuristic algorithms and the optimal solutions. Lastly, Section 6 concludes the whole research.

2. The Definition of our PBAR Problem

In this section, we will introduce some assumptions, notations, and definitions.

2.1 Assumptions

The following states some important assumptions used in our research.

(1) We assume that the ad hoc network’s topology would not change, i.e. no host gets move [5, 7].

(2) We only consider the transmission power and ignore the reception power [2].

(3) We assume that when a data packet passes through a link, the transmission power consumption associated with the link can be an arbitrary value, i.e., can be independent of the Euclidean length of link [2].

2.2 Problem Formulation

We represent an ad hoc network by a weighted graph G = (V, E), where V denotes the set of hosts (including source hosts, destination hosts, and intermediating hosts) and E denotes the set of communication channels connecting the hosts. Let

1 2

{ , ,..., }

n

d d d

D= v v v be a set of destination hosts. For D,

we define a data packet requirement function γ: D→I+.

The value γ( )vd_i represents the number of data packets

required by destination . For E, we define a transmission power consumption function

i

d

v

β

: →

that assigns a nonnegative weight to each link in the network. The value associated with link

E

+

R

( , )v vi j β

( , )v vi j ∈E represents the transmission power that node

will consume when one data packet is delivered through that link. For E, we define a packet flow function

i

v

:

f E→I+. The value denotes the number of

data packets passing through link . For V, we define a node power consumption function

( , )_i _j f v v ( , )v v_i _j :V R α _→ +_. Thus, ( , ) ( ) ( , ) ( , ) i j i i j v v E v f v v v α ∈

=

∑

×β i vj represents the total

transmission power that node will consume during the deliveries of all the data packets required by all the destination nodes.

i

v

Based on these notations and definitions, now we can formally describe our power-balanced anycast routing (PBAR) problem as follows: given a weighted graph G=(V,E), a set of source hosts S={ ,vs₁ vs₂,...,vs_m}

and a set of destination hosts , a

data packet requirement function

1 2 { , ,..., } n d d d D= v v v : D I γ _→ + _{, a}

transmission power consumption function β : → ,

find a set of routing paths such that (1) the data packet requirement function of each destination node is satisfied and (2) the maximum of node’s transmission power

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consumption in

G

is minimized, i.e., max

{

( )

}

i i v G∈ α v is minimized.

As an example, let us consider Figure 1, where an ad hoc network is shown with three source nodes and two destination nodes. The number next to each node represents the number of data packets required by the node. The number next to each link represents the power to be consumed when one data packet is delivered through the link. Figure 1 shows a set of routing paths with the best power balance. In this case, the maximum of node’s transmission power consumption in the network _max

{

_{( )}

}

i

v G∈ α v is equal to α( )vs2 = 4 + 4 = 8,

which is the best solution.

3. The Complexity of our PBAR Problem

In this section, we will show that our PBAR problem is NP-complete. To prove our PBAR problem to be NP-complete, first let us restate it in its decision version as follows: given a weighted graph G=(V,E), a set of source hosts S={ ,vs₁ vs₂,...,vs_m} , a set of

destination hosts , a data packet

requirement function , a transmission power

consumption function 1 2 { , ,..., } n d d d D= v v v : D I γ _→ +

β

: E → R+ , a

power-constrained constant , find a set of routing

paths such that (1) the data packet requirement function of each destination node is satisfied and (2) the maximum of node’s transmission power consumption in

is less than or equal to , i.e., p

c

G

c

p max

{

( )

}

i i v G∈ α v

≤

. p

c

For simplicity, in what follows, we will not distinguish the decision version and the optimal version of the PBAR problem when no ambiguity arises.

Next, let us introduce the 3-Dimensional Matching (3DM) problem [3].

Instance: A set M

⊆

, where W, X, and Y are disjoint sets having the same number q of elements.

W

× ×

X

Y

Question: Does M contain a matching, that is, a subset

M ′

⊆

M such that

M ′

= q and no two elements of

M ′

agree in any coordinate?

This problem was shown to be NP-complete by Karp [3]. Now, we will use it to prove the following theorem.

Theorem 1. The PBAR problem is NP-complete. Proof. First, the PBAR problem can be easily seen to be

in the class NP. We next transform the 3DM problem to the PBAR problem in polynomial time. Let the sets W, X,

Y, with

W

=

X

=

Y

=

q

, and

M

⊆

W

×

X

×

Y

be an arbitrary instance of 3DM. Let the elements of these sets be denoted by

{

1, , ,2 q

}

W= w w w ,X=

{

x x₁, , ,₂ x_q

}

{

1 2

}

,Y= y y, , ,y_q and

M

=

{

m

1

,

m

2

,

m

k

}

, where

M

k= . We construct an instance of the PBAR problem

as follows: For each element

w

_i (

x

_i, _i) of W, the corresponding weighted graph G = (V, E) has a source node (an intermediate nodes , a destination

node ) (1

y

i w

v

i x

v

i y

v

≤

i

≤

q). Thus, V = { } { 1 2 xq } { y1 y2 yq }. If ( , 1

,

2

, ,

q w w w

v

…

v

∪

,

, ,

x x

v

…

v

∪

v

,

v

, ,

…

v

i

w

x

_j ,

y

_k )

∈

M, then there exist one edge

i j w

,

x

v

<

>

between nodes i w and

v

j x , and one edge j k x y

v

,

v

<

>

between nodes j x

v

and . Thus,

the edge set E = { : if

( , k y

v

,

i j w x

v

<

>

i

w

x

_j ,

y

_k )

∈

M } {∪ j k x y

<

>

: if ( _i , _j

,

v

w

x

,

y

_k)

∈

M }. The number of data packets

required by each destination node

k y is assumed to be k y

v

(v ) 1

γ

= . Each edge has a transmission power consumption of 1 when one data packet traverses it. Finally, let _p=1. The constructed G is illustrated in Figure 2. It is easy to see that this transformation can be finished in polynomial time.

c

We next show that there exists a set of feasible routing paths for the PBAR problem in G if and only if the set M contains a matching

M ′

. First, suppose M contains a matching, that is, a subset M ′

⊆

M such that

M ′

= q and no two elements of

M ′

agree in any coordinate. If

(

w

_i

′

,

x

′

_j

,

y

_k

′

)

∈

M

′

, then we let

(

, ,

)

i k

w x_j y v ′ v′ v ′

w

v

be a routing path in G between source

node

i′ and destination node vyk′. Since

M

′

=

q

,

there are q routing paths each of which is for a different pair of source and destination nodes. Since no two elements of

M ′

agree in any coordinate, these routing paths are pairwise node-disjoint. Because they are pairwise node-disjoint, any link ( _i, _j) belongs to at

most one of these q paths. Therefore, for each link

v v

( , )v vi j ∈E , f v v( , ) 1i j ≤ . Similarly, any node

belongs to at most one of these q paths. So, for any i i

v

v ∈V, ( , ) ( , ) 1 i j i j v v E f v v ∈ ≤

∑

. As a result, for any vi∈V, we have ( , )v vi j∈E ( )vi ( , )v vi j f v v( , )i α =

∑

β × j ( , )v vi j E 1 f v v( , ) 1i j cp =

∑

× ≤ = ∈ . Thus, these q routing paths form a set of feasible routing paths for the corresponding PBAR problem in G.

Next, suppose we have a solution for the PBAR problem in the weighted graph G. Let P P1, 2, ,Pq be

one of the possible solutions in G . Because

( )

v

i

c

p

1 α

≤

=

for each node

v

_iand each link has a transmission power consumption of 1, each node _i belongs to at most one path

v

l

P (otherwise,

( )

v

i

c

p

1 α

>

=

). Thus paths P P1, 2, ,Pq are

pairwise node-disjoint. If Pt∈

(

P₁, , ,P₂ Pq

)

, where

( , ,

ti t t )

t w x

P = v v v

j yk , then let

(

w xti, tj,ytk

)

∈M ′.

Clearly M ′= q, M ′

⊆

M, and no two elements of M ′

agree in any coordinate. Thus, M ′ is a matching. This

completes our proof of NP-completeness. ▓

4. An Efficient GA for the PBAR problem

In this section, we will develop an efficient GA to solve the PBAR problem according to the principle of GAs.

4.1 Representation of Chromosomes

Given an instance of the PBAR problem, where the given graph is G = (V, E), N = |V|. Let

1

,

2

, ,

m

s s s

be a set of source nodes,

n

d be a set of destination nodes, and be other nodes.

v v

v

,

, ,

d d

v

1

,

2

, ,

h o o o

v

1 2

v

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Without loss of generality, the nodes in G is numbered from v1 to , and 1 2 m 1 h 1 n N

v

s <s < <s < <o <o <d < < d . A

possible solution (i.e., a set of routing paths each of which is dedicated to one data packet) is represented by a chromosome c, which is a string of genes with length

P N, where P denotes the total number of data packets

required by all the destinations. To facilitate the following presentation, a chromosome will be depicted as a P N matrix each of whose rows exactly

corresponds to one routing path. We will use

d

×

j k v

R

to denote the row corresponding to the routing path dedicated to the kth data packet required by destination node

j

d , and use

C

z to denote the column corresponding to node , where z

v

z

v

∈

{

s s

₁

, , , , , , ,

₂

s o

m ₁

o d

h ₁

, ,

d

n

}

. The columns

are arranged by their indexes. 1

s

C

is the first column and is the last column (see Figure 3).

n

d

A gene is a bit and gene g

C

i being 1 (0) means its corresponding vertex vi being (being not) included in the constructed routing path. Because any data packet must be from exact one source node, exact one of the first m genes within a row should always be set to 1. Furthermore, because a row

d j

k v

R is corresponding to the kth data packet required by destination node

j

d , its

j gene should be set to 1. Therefore, at some places in our GA, we will be only concerned with the genes in columns

1

o within a row. A gene in column

z, where z

v

( )thd C C

∈

2

{

, o ,Co_h

C

o

₁

, ,

o

h

}

and in row _d j k v

R

being 1 means the associated routing path destinated for

j

d must pass through node _z . Finally, to decode a chromosome into a set of routing paths, our decoding method is to establish a shortest path

d

v

j k v

p

between source i s

v

and destination j

d from each row _d

v

j

k v

R

with its gene being 1 and gene being 1 by Dijkstra’s algorithm. As an example, given the graph in Figure 1, the set of routing paths induced by the chromosome in Figure 3(a) is shown in Figure 3(b).

( )th

s

i ( )thdj

4.2 Fitness Function

Obviously, under such a chromosome’s representation, a set of routing paths will be induced by each chromosome, which is generated at random and may experience the crossover and mutation operations. For a chromosome, the associated power consumption of each node will depend on the set of resulted routing paths. To measure the quality of a chromosome, we need to define a fitness function such that when the maximal node’s transmission power consumption _max

_{

_{( )}

_}

i

i v G∈ α v

is smaller, the corresponding chromosome will be assigned a higher fitness value. In our GA, the fitness value of a chromosome is the reverse of the maximal node’s transmission power consumption _{1/ max}

{

_{( )}

}

i i

v∈G α v associated with the set of routing paths contained in the chromosome. For example, the fitness value of the chromosome shown in Figure 3(a) is

1

1/ ( ) 1/(5 8) 1/13α vs = + = . Given a chromosome ci, the

fitness function of ci can be computed by Dijkstra’s

algorithm.

4.3 The Detailed Procedure of our GA

In the following we will explain each step in our GA

Step 1. Initialization of chromosomes: Step 1 generates

np different chromosomes at random, which form the first generation of chromosomes. In a chromosome, because each row is exactly corresponding to one data packet, i.e., a row

d j is corresponding to the kth data packet

required by destination node k

v

R

j

d , its gene will be definitely set to 1. Because any data packet must be from exact one source node, we will randomly select exact one of the first m genes within a row to set its value to 1. All the genes in columns within a row will initially be set to 0.

v

( )thd_j

1

,

2

,

h

o o o

C C

C

Step 2. Evaluation of chromosomes: At step 2, we

compute the fitness value of each chromosome using Dijkstra’s algorithm according to Subsection 4.2.

Step 3. Termination criteria: Because the PBAR

problem is a NP-complete problem, its optimal solutions are very hard to find. Therefore, we cannot use the gap between the optimal solution and the current suboptimal solution to determine when to stop our GA. Instead, we adopt a common terminal rule, the maximum generation number MAX_ng, to terminate the evolution of our GA. Step 4. Duplication: At this step, we make one copy

g

n

S ′

of the current generation of chromosomes. This copy will be used at step 8.

Step 5. Selection: According to the computed fitness

values, some of chromosomes are selected in order to generate more offspring through crossover and mutation operations, and others are removed from the chromosome pool. Note that the number of the chromosomes in each generation is always restricted to

np. The probability of any chromosome ci to be selected from the population is defined as

_{( )}

n

_{( )}

1 j / p i j f c

∑

f c =

where is the fitness function of chromosome ,

)

(

c

i

f

i. This selection is implemented by using a routette wheel selection scheme [4].

c

Step 6. Perform crossover operation on the selected chromosomes: In crossover, we select a pair of

chromosomes and do crossover operation on them to exchange information between them. In crossover, a crossover rate rc is first given (the method to determine a proper value of rc will be discussed in detail in Section 5). For each chromosome, a random number between 0 and 1 is generated. If the random number is less than the given crossover rate, the chromosome will be marked to indicate that crossover will be executed with it. When all the chromosomes have finished the mark operation, we will in sequence select a pair of marked chromosomes to perform crossover on them. Our GA adopts one-cut-point crossover [4] in which the two selected chromosomes exchange their genes according to the selected rows. Figure 4 shows how the crossover operation is performed on a pair of chromosomes.

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Step 7. Perform mutation operation on the selected genes: The purpose of mutation is to generate a variety

of chromosomes to avoid the local optimal solution. Given a mutation rate rm (which is decided by simulations). Because any data packet must be from exact one source node and to exact one destination node, the mutation operation is limited on genes whose indexes belong to

{

o

₁

, ,

o

h

}

. In reality, a random number between 0 and 1 is generated for each gene in columns

1 2 h

o o o . If the random number is less than the given mutation rate, then the corresponding gene _z will do a mutation, i.e., the value of the gene will be inversed. If _z = 1, this implies the routing path associated with the row must include node .

, ,

C C C

g

z

Step 8. Reproduction: Our reproduction operation will

generate the next generation of chromosomes by picking up the top 50% chromosomes with higher fitness from the current generation

g

n of chromosomes and the set

g

n of chromosomes, which is obtained from n_g

v

S ′

S ′′

S ′

through the selection, crossover, and mutation operations.

5. Computer Simulations

In this section, by means of computer simulations, we will examine the efficiency of our GA and compare the solutions generated by our GA with those generated by two other traditional heuristic algorithms, which are called the SPAR algorithm and the PBAR algorithm, respectively [6]. The SPAR algorithm is based on the shortest path strategy while the PBAR algorithm has already adopted the power-balanced concept.

5.1 Determining the proper values for different parameters in our GA

Like any other GAs, in order to yield the best performance, the four main parameters in our GA: the size _p of population, the crossover rate _c , the mutation rate , and the maximum number of generations

ng

n

r

m

r

MAX , must be properly selected.

According to the simulation results in [6], it can be found that the more proper values for these parameters are as follows: n_p=50 , r_c =0.7 ,

r

m

=

0.07

, . 50 ng 5.2 Simulation Results MAX = 4

In this subsection, we will present and discuss the simulation results of our GA, the SPAR algorithm, and the PBAR algorithm in two different simulation environments. We will observe the maximal node’s power consumption in the network in our simulations.

The environments in our first simulation are set as follows: the network consists of 50 nodes which are located in a 100 100 m

×

2_{area randomly. The number of}

links is set to be , where N is 50. The transmission power consumption of each link is set to be a value between 10 and 40 randomly. The number of destination nodes is set to be 10. The number of data packets required by each destination node is set to be a value between 1 and 5 randomly. In our first simulation environment, we will observe how the number of source nodes impacts on the performances by our GA and the

two traditional heuristic algorithms. Figure 5 shows the simulation results, where we vary the number of source nodes from 10 to 30. From Figure 5, it can be found that the maximal node’s power transmission consumption in the network produced by our GA is much lower than those produced by the two other heuristic algorithms. We can also observe that the maximal node’s transmission power consumption decreases with the raising of the number of source nodes. This is because the more source nodes exist, there are more chances for anycast routing algorithms to select the proper source nodes to send data packets to each destination node.

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N× N−

The environments in our second simulation are the same as those in our first simulation except that the number of source nodes is set to be 15 and the number of links becomes adjustable. In our second simulation environment, we will observe how the total number of links impacts on the performances of the three heuristic algorithms. From Figure 6 (where N denotes the total number of nodes in the network, and in this case N is 50), it can be easily found that the maximal node’s transmission power consumption of the three heuristic algorithms decrease with the raising of the total number of links. This is because the more the total number of links is, there are more chances for the anycast routing algorithms to select the proper routing paths to route data packets to each destination node. The maximal node’s transmission power consumption obtained by our GA is also lower than the values obtained by the other two traditional heuristic algorithms.

6. Conclusions

In this paper, we have discussed the PBAR problem and proved it to be NP-complete. Based on the principle of GAs, in this paper, we have designed an efficient heuristic algorithm for the PBAR problem. Computer simulations verify that the power consumptions generated by our genetic power-balanced anycast routing algorithm are more balanced than those generated by shortest-path-based anycast routing algorithms and other traditional power-balanced anycast routing algorithms.

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[9] S. Singh, M. Woo, and C. S. Raghavendra, "Power-Aware Routing In Mobile Ad Hoc Networks," Proc. MobiCom, 1998.

v_s1 v_s2 v_s3 v_o1 v_o2 _v o3 v_d1 v_d2 5 β = β =5 4 β = β =4 β =3 _{β =}₆ 8 β = 4 β = 2 γ= γ=2 6 β = 6 β = β =10 2 ( ) 4 4 8vs α = + =

Figure 1: An illustration of our PBAR problem.

vw1 vx1 vy1 vw2 vx2 vy2 vw₃ vx₃ vy₃ vwq-1 vxq-1 vyq-1 vw_q vx_q vy_q . . . . . . . . . 1 β = β =1 1 β = β =1 1 β = β =1 1 β = β =1 1 β = β =1 1 β = β =1 β =1 β =1 1 β = β =1 β =1 β =1 1 β = β =1 β =1 β =1 1 γ= 1 γ= 1 γ= 1 γ= 1 γ= p

c

=1

Figure 2: An illustration of Theorem 1.

C_s1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 C_s2 C_s3 C_o1 C_o2 C_o3 C_d1 C_d2 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R (a) a chromosome v s1 vs2 vs3 v o1 vo2 v_o3 v d1 vd2 5 β = β =5 4 β = 6 β = 4 β = β =3 8 β = 4 β = 2 γ= γ=₂ 6 β = 6 β = β =10 1 ( ) 5 8 13vs α = + = 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R

(b) the set of routing paths corresponding to the chromosome in (a)

Figure 3: Representation of chromosomes.

crossover point 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 Cs1Cs2Cs3Co1Co2Co3Cd1Cd2 Cs1Cs2Cs3Co1Co2Co3Cd1Cd2 Cs1Cs2Cs3Co1Co2Co3Cd1Cd2 Cs1Cs2Cs3Co1Co2Co3Cd1Cd2 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R 1 1 d v R 1 2 d v R 2 1 d v R 2 2 d v R

Figure 4: Crossover operation.

0 20 40 60 80 100 120 140 160 10 15 20 25 30 the number of source nodes

th e m axi m al node ’s tr an sm is iion pow er consum pt ion i n t he ne tw or k GA PBAR SPAR

Figure 5: The influence of the number of source nodes. 40 50 60 70 80 90 100 110 120 130 5×N 10×N 15×N 20×N 25×N total number of links (N=50)

th e m ax im al n od e’ s tr an sm is iio n po w er co ns um ptio n in th e ne tw or k GA PBAR SPAR